Bookmark

Mathematics > Dynamical Systems

Title:Counting periodic trajectories of Finsler billiards

Abstract: We provide lower bounds on the number of periodic Finsler billiard
trajectories inside a quadratically convex smooth closed hypersurface M in a
d-dimensional Finsler space with possibly irreversible Finsler metric. An
example of such a system is a billiard in a sufficiently weak magnetic field.
The r-periodic Finsler billiard trajectories correspond to r-gons inscribed in
M and having extremal Finsler length. The cyclic group Z/rZ acts on these
extremal polygons, and one counts the Z/rZ-orbits. Using Morse and
Lusternik-Schnirelmann theories, we prove that if r is an odd prime, then the
number of r-periodic Finsler billiard trajectories is not less than
(r-1)(d-2)+1. We also give stronger lower bounds when M is in general position.
The problem of estimating the number of periodic billiard trajectories from
below goes back to Birkhoff. Our work extends to the Finsler setting the
results previously obtained for Euclidean billiards by Babenko, Farber,
Tabachnikov, and Karasev.